Best Known (72, 72+14, s)-Nets in Base 7
(72, 72+14, 117651)-Net over F7 — Constructive and digital
Digital (72, 86, 117651)-net over F7, using
- net defined by OOA [i] based on linear OOA(786, 117651, F7, 14, 14) (dual of [(117651, 14), 1647028, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(786, 823557, F7, 14) (dual of [823557, 823471, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(786, 823558, F7, 14) (dual of [823558, 823472, 15]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(71, 15, F7, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(786, 823558, F7, 14) (dual of [823558, 823472, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(786, 823557, F7, 14) (dual of [823557, 823471, 15]-code), using
(72, 72+14, 823559)-Net over F7 — Digital
Digital (72, 86, 823559)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(786, 823559, F7, 14) (dual of [823559, 823473, 15]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(715, 16, F7, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,7)), using
- dual of repetition code with length 16 [i]
- linear OA(71, 16, F7, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(11) [i] based on
(72, 72+14, large)-Net in Base 7 — Upper bound on s
There is no (72, 86, large)-net in base 7, because
- 12 times m-reduction [i] would yield (72, 74, large)-net in base 7, but